Calculus Examples

$\int_{π}^{3 π} \cot^{2} (\frac{x}{6}) d x$

Step 1

Let $u = \frac{x}{6}$ . Then $d u = \frac{1}{6} d x$ , so $6 d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = \frac{x}{6}$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $\frac{x}{6}$ .

$\frac{d}{d x} [\frac{x}{6}]$

Step 1.1.2

Since $\frac{1}{6}$ is constant with respect to $x$ , the derivative of $\frac{x}{6}$ with respect to $x$ is $\frac{1}{6} \frac{d}{d x} [x]$ .

$\frac{1}{6} \frac{d}{d x} [x]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{1}{6} \cdot 1$

Step 1.1.4

Multiply $\frac{1}{6}$ by $1$ .

$\frac{1}{6}$

Step 1.2

Substitute the lower limit in for $x$ in $u = \frac{x}{6}$ .

$u_{lower} = \frac{π}{6}$

Step 1.3

Substitute the upper limit in for $x$ in $u = \frac{x}{6}$ .

$u_{upper} = \frac{3 π}{6}$

Step 1.4

Cancel the common factor of $3$ and $6$ .

Tap for more steps...

Step 1.4.1

Factor $3$ out of $3 π$ .

$u_{upper} = \frac{3 (π)}{6}$

Step 1.4.2

Cancel the common factors.

Tap for more steps...

Step 1.4.2.1

Factor $3$ out of $6$ .

$u_{upper} = \frac{3 π}{3 \cdot 2}$

Step 1.4.2.2

Cancel the common factor.

$u_{upper} = \frac{3 π}{3 \cdot 2}$

Step 1.4.2.3

Rewrite the expression.

$u_{upper} = \frac{π}{2}$

Step 1.5

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = \frac{π}{6}$

$u_{upper} = \frac{π}{2}$

Step 1.6

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{\frac{π}{6}}^{\frac{π}{2}} \cot^{2} (u) \frac{1}{\frac{1}{6}} d u$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{6}$ .

$\int_{\frac{π}{6}}^{\frac{π}{2}} \cot^{2} (u) (1 \cdot 6) d u$

Step 2.2

Multiply $6$ by $1$ .

$\int_{\frac{π}{6}}^{\frac{π}{2}} \cot^{2} (u) \cdot 6 d u$

Step 2.3

Move $6$ to the left of $\cot^{2} (u)$ .

$\int_{\frac{π}{6}}^{\frac{π}{2}} 6 \cot^{2} (u) d u$

Step 3

Since $6$ is constant with respect to $u$ , move $6$ out of the integral.

$6 \int_{\frac{π}{6}}^{\frac{π}{2}} \cot^{2} (u) d u$

Step 4

Using the Pythagorean Identity, rewrite $\cot^{2} (u)$ as $- 1 + \csc^{2} (u)$ .

$6 \int_{\frac{π}{6}}^{\frac{π}{2}} - 1 + \csc^{2} (u) d u$

Step 5

Split the single integral into multiple integrals.

$6 (\int_{\frac{π}{6}}^{\frac{π}{2}} - 1 d u + \int_{\frac{π}{6}}^{\frac{π}{2}} \csc^{2} (u) d u)$

Step 6

Apply the constant rule.

$6 (- u]_{\frac{π}{6}}^{\frac{π}{2}} + \int_{\frac{π}{6}}^{\frac{π}{2}} \csc^{2} (u) d u)$

Step 7

Since the derivative of $- \cot (u)$ is $\csc^{2} (u)$ , the integral of $\csc^{2} (u)$ is $- \cot (u)$ .

$6 (- u]_{\frac{π}{6}}^{\frac{π}{2}} + - \cot (u)]_{\frac{π}{6}}^{\frac{π}{2}})$

Step 8

Combine $- u]_{\frac{π}{6}}^{\frac{π}{2}}$ and $- \cot (u)]_{\frac{π}{6}}^{\frac{π}{2}}$ .

$6 (- u - \cot (u)]_{\frac{π}{6}}^{\frac{π}{2}})$

Step 9

Evaluate $- u - \cot (u)$ at $\frac{π}{2}$ and at $\frac{π}{6}$ .

$6 (- \frac{π}{2} - \cot (\frac{π}{2}) - (- \frac{π}{6} - \cot (\frac{π}{6})))$

Step 10

Simplify.

Tap for more steps...

Step 10.1

The exact value of $\cot (\frac{π}{2})$ is $0$ .

$6 (- \frac{π}{2} - 0 - (- \frac{π}{6} - \cot (\frac{π}{6})))$

Step 10.2

The exact value of $\cot (\frac{π}{6})$ is $\sqrt{3}$ .

$6 (- \frac{π}{2} - 0 - (- \frac{π}{6} - \sqrt{3}))$

Step 10.3

Multiply $- 1$ by $0$ .

$6 (- \frac{π}{2} + 0 - (- \frac{π}{6} - \sqrt{3}))$

Step 10.4

Add $- \frac{π}{2}$ and $0$ .

$6 (- \frac{π}{2} - (- \frac{π}{6} - \sqrt{3}))$

Step 10.5

To write $- (- \frac{π}{6} - \sqrt{3})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$6 (- \frac{π}{2} - (- \frac{π}{6} - \sqrt{3}) \cdot \frac{2}{2})$

Step 10.6

Combine $- (- \frac{π}{6} - \sqrt{3})$ and $\frac{2}{2}$ .

$6 (- \frac{π}{2} + \frac{- (- \frac{π}{6} - \sqrt{3}) \cdot 2}{2})$

Step 10.7

Combine the numerators over the common denominator.

$6 \frac{- π - (- \frac{π}{6} - \sqrt{3}) \cdot 2}{2}$

Step 10.8

Multiply $2$ by $- 1$ .

$6 \frac{- π - 2 (- \frac{π}{6} - \sqrt{3})}{2}$

Step 10.9

Combine $6$ and $\frac{- π - 2 (- \frac{π}{6} - \sqrt{3})}{2}$ .

$\frac{6 (- π - 2 (- \frac{π}{6} - \sqrt{3}))}{2}$

Step 10.10

Cancel the common factor of $6$ and $2$ .

Tap for more steps...

Step 10.10.1

Factor $2$ out of $6 (- π - 2 (- \frac{π}{6} - \sqrt{3}))$ .

$\frac{2 (3 (- π - 2 (- \frac{π}{6} - \sqrt{3})))}{2}$

Step 10.10.2

Cancel the common factors.

Tap for more steps...

Step 10.10.2.1

Factor $2$ out of $2$ .

$\frac{2 (3 (- π - 2 (- \frac{π}{6} - \sqrt{3})))}{2 (1)}$

Step 10.10.2.2

Cancel the common factor.

$\frac{2 (3 (- π - 2 (- \frac{π}{6} - \sqrt{3})))}{2 \cdot 1}$

Step 10.10.2.3

Rewrite the expression.

$\frac{3 (- π - 2 (- \frac{π}{6} - \sqrt{3}))}{1}$

Step 10.10.2.4

Divide $3 (- π - 2 (- \frac{π}{6} - \sqrt{3}))$ by $1$ .

$3 (- π - 2 (- \frac{π}{6} - \sqrt{3}))$

Step 11

Simplify.

Tap for more steps...

Step 11.1

Apply the distributive property.

$3 (- π - 2 (- \frac{π}{6}) - 2 (- \sqrt{3}))$

Step 11.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.2.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

$3 (- π - 2 \frac{- π}{6} - 2 (- \sqrt{3}))$

Step 11.2.2

Factor $2$ out of $- 2$ .

$3 (- π + 2 (- 1) \frac{- π}{6} - 2 (- \sqrt{3}))$

Step 11.2.3

Factor $2$ out of $6$ .

$3 (- π + 2 \cdot - 1 \frac{- π}{2 \cdot 3} - 2 (- \sqrt{3}))$

Step 11.2.4

Cancel the common factor.

$3 (- π + 2 \cdot - 1 \frac{- π}{2 \cdot 3} - 2 (- \sqrt{3}))$

Step 11.2.5

Rewrite the expression.

$3 (- π - \frac{- π}{3} - 2 (- \sqrt{3}))$

Step 11.3

Multiply $- 1$ by $- 2$ .

$3 (- π - \frac{- π}{3} + 2 \sqrt{3})$

Step 11.4

Move the negative in front of the fraction.

$3 (- π - - \frac{π}{3} + 2 \sqrt{3})$

Step 11.5

Multiply $- - \frac{π}{3}$ .

Tap for more steps...

Step 11.5.1

Multiply $- 1$ by $- 1$ .

$3 (- π + 1 \frac{π}{3} + 2 \sqrt{3})$

Step 11.5.2

Multiply $\frac{π}{3}$ by $1$ .

$3 (- π + \frac{π}{3} + 2 \sqrt{3})$

Step 11.6

To write $- π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 (- π \cdot \frac{3}{3} + \frac{π}{3} + 2 \sqrt{3})$

Step 11.7

Combine $- π$ and $\frac{3}{3}$ .

$3 (\frac{- π \cdot 3}{3} + \frac{π}{3} + 2 \sqrt{3})$

Step 11.8

Combine the numerators over the common denominator.

$3 (\frac{- π \cdot 3 + π}{3} + 2 \sqrt{3})$

Step 11.9

Multiply $3$ by $- 1$ .

$3 (\frac{- 3 π + π}{3} + 2 \sqrt{3})$

Step 11.10

Add $- 3 π$ and $π$ .

$3 (\frac{- 2 π}{3} + 2 \sqrt{3})$

Step 11.11

Move the negative in front of the fraction.

$3 (- \frac{2 π}{3} + 2 \sqrt{3})$

Step 11.12

Apply the distributive property.

$3 (- \frac{2 π}{3}) + 3 (2 \sqrt{3})$

Step 11.13

Cancel the common factor of $3$ .

Tap for more steps...

Step 11.13.1

Move the leading negative in $- \frac{2 π}{3}$ into the numerator.

$3 \frac{- 2 π}{3} + 3 (2 \sqrt{3})$

Step 11.13.2

Cancel the common factor.

$3 \frac{- 2 π}{3} + 3 (2 \sqrt{3})$

Step 11.13.3

Rewrite the expression.

$- 2 π + 3 (2 \sqrt{3})$

Step 11.14

Multiply $2$ by $3$ .

$- 2 π + 6 \sqrt{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$- 2 π + 6 \sqrt{3}$

Decimal Form:

$4.10911953 \dots$